Skill: curve-sketching

Curve Sketching Practice

Read a graph from its derivatives: f′ tells you where the curve climbs and falls, f″ tells you how it bends. Find critical points, classify them, locate inflection points, and sketch with confidence.

1 Practice

Type your answer and press Check answer (or Enter). A wrong answer earns a hint; a second miss earns a stronger hint; a third miss walks you through the full solution. Scores and streaks are session-only.

Score 0 · Streak 0 · Problem 0

Score counts first-try correct answers. Streak counts consecutive correct answers.

Problem

Enter integers or decimals. For two critical numbers, enter the smaller first. Multiple choice: click an option.

2 What you’ll practice

Every problem is generated fresh from templates across several question types — a problem never repeats until its whole pool is used up.

Critical points

Solve f′(x) = 0 for cubics with integer critical numbers.

Ex: f(x) = x³−3x²−9x+12 → x = −1, 3

Classify critical points

Second-derivative test — including the “neither” trap.

Ex: f(x) = x³, classify x = 0

Inflection points

Where f″ = 0 with a genuine sign change.

Ex: f(x) = x³+3x² → inflection at x = −1

Increase / decrease

Find where a parabola switches direction.

Ex: f(x) = x²−4x → switch at x = 2

Concavity intervals

Solve f″(x) > 0 as an inequality.

Ex: concave up on (L, ∞) → L = ?

Read the sign chart

From f′ signs to max / min / neither.

Ex: + then − at x = 2 → ?

Second-derivative test

Classify from f′(c) = 0 and the sign of f″(c).

Ex: f′(3)=0, f″(3)=−4 → ?

3 Watch out for these

The mistakes students make most often on these problems.

Every critical point is a max/min

Test each one: f(x) = x³ has f′(0) = 0 but no extremum — just a flat inflection spot.

Concavity = increasing

f′ controls up/down; f″ controls the bend. A decreasing function can be concave up.

f″ = 0 means inflection

Only with a sign change of f″. f(x) = x⁴ has f″(0) = 0 and no inflection.

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